Ray shooting and parametric search
Efficient algorithms for the ray shooting problem are presented: Given a collection Γ of objects in Rd, build a data structure so that, for a query ray, the first object of Γ hit by the ray can be quickly determined. Using the parametric search technique, this problem is reduced to the segment emptiness problem. For various ray shooting problems, space/query-time trade-offs of the following type are achieved: For some integer b and a parameter m (n≤m≤nb) the queries are answered in time O((n/m1/b)logO(1)n), with O(m1+ε) space and preprocessing time (ε>0 is arbitrarily small but fixed constant). b = [d/2] is obtained for ray shooting in a convex d-polytope defined as an intersection of n half spaces, b = d for an arrangement of n hyperplanes in Rd, and b = 3 for an arrangement of n half planes in R3. This approach also yields fast procedures for finding the first k objects hit by a query ray, for searching nearest and farthest neighbors, and for the hidden surface removal. All the data structures can be maintained dynamically in amortized time O(m1+ε/n) per insert/delete operation.
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